The description of crystal surfaces requires some basic knowledge of crystallography. Therefore, this chapter presents a short overview of crystal lattices and their classification due to symmetry. This knowledge is required to understand the substrate structure and the orientation of the surface. However, the 3-D point groups, space groups and the mathematical description of symmetry operations in three dimensions are not described here: for a more detailed explanation the reader is referred to the International Tables of Crystallography [2.1], which is the standard reference book, or a number of textbooks on crystallography published by the International Union of Crystallography [2.2–2.5]. The 2-D space groups and symmetry operations are explained with somewhat more detail here because these are frequently used in surface structure determination. A very detailed description of the geometry of crystal surfaces is given in a recent book by K. Hermann [2.6]. A short introduction into the kinematic theory of diffraction and into diffraction at 2-D periodic lattices is also included here.

We discuss here the methods of quantitative LEED I(V) analysis and their application to relatively complex types of surface structures: quasicrystalline and modulated surfaces.

X-ray crystallography of 3-D bulk materials opened entire new fields of discovery in the twentieth century, from elemental crystals to DNA molecules. Similarly, the determination of atomic-scale structure of 2-D surfaces of condensed matter has achieved fundamental new understanding and generated powerful techniques that helped spawn new areas of research, from catalysis to nanotechnology, for the twenty-first century. Specific examples include, among many others: new catalysts; various new carbon structures (such as buckminsterfullerenes, nanotubes and graphene); quantum dots used in optoelectronic displays (including television displays); molecules allowing electron transport and switching; nanoparticles enabling targeted drug delivery; and nanomachines for future manufacturing and medical applications.

LEED has found widespread application in surface science, since the LEED experiment can be performed in a small laboratory and LEED systems are commercially available. A main advantage compared to surface X-ray diffraction is that on the LEED screen most of the 2-D diffraction pattern is visible, thus allowing a quick and comprehensive overview of the symmetry and to some extent about the degree of ordering of the surface under examination. A LEED system is therefore included in most UHV chambers to control the quality of the surface preparation for a wide range of surface studies. A qualitative interpretation of the diffraction pattern is the most common use of LEED: it allows the identification of the surface unit cell, the estimation of the degree of ordering and the identification of different surface phases in adsorption systems (and thereby often a check on adsorbate coverage). The diffraction pattern thus reflects the translational symmetry and the crystalline order of the surface.